Problem: Solve for $r$: $\frac{r+9}{r-3} = \frac{r-2}{r+5}$
Explanation: Cross-multiplying (which is the same as multiplying both sides by $r-3$ and by $r+5$) gives \[(r+9)(r+5) = (r-2)(r-3).\]Expanding the products on both sides gives  \[r^2 + 9r + 5r + 45 = r^2 -2r - 3r + 6.\]Simplifying both sides gives $r^2 + 14r + 45 = r^2 - 5r + 6$.  Simplifying this equation gives $19r = -39$, so $r = \boxed{-\frac{39}{19}}$.